Suppose one has some capital to invest in the stock market. How does one go about investing it? One can try picking “good” stocks at low prices and selling them later at higher prices. That age old strategy although quite simple conceptually, is very difficult to implement. Stocks are inherently risky since they move up and down in a seemingly haphazard fashion on a variety of inputs and the common wisdom says that one should not keep “all eggs in one basket”, but rather spread out the investment so as to minimize the risk.
In the 1950s Harry Markowitz, then a graduate student at the University of Chicago, fine-tuned this idea, laying the foundations of the modern portfolio theory. Markowitz's idea is easy to understand. Let us concentrate on picking stocks. The strategy is to pick the right mix of stocks that minimizes the overall risk in terms of losing money that is invariantly caused by the stock values moving below their purchased prices. Stocks move up and down, sometimes violently, causing great volatility in term of the total portfolio value. Markowitz's basic idea was to keep this volatility low by picking the right mix of stocks. One would like to keep the total portfolio value fluctuations to a minimum at all times, i.e. no big variations, and if there are any variations they should amount to small jitters. Actual stock variations are of course beyond one's control, but what is controllable is which specific stocks to add to the overall portfolio from the total pool, and how much of each stock. The idea is to use the right mix of right stocks to minimize the overall volatility. After all the basic goal of a fund manager is to protect the portfolios under his management from losing their values and hopefully increase their return values or the overall gain. The specific stock holdings and their relative importance within the portfolio are unimportant both to the fund manager and to the investor, so long as the portfolio “makes money”, or performs well.
Thus two quantities play a role in portfolio selection—the overall risk, and the overall return or gain. Obviously, the overall risk needs to be minimized, and the overall return or gain should be maximized at the same time. Various strategies can be designed using these conflicting goals.
For a random variable, the variance is a good measure of the spread of the random variable around a mean value, and hence volatility minimization for stocks or investments, can be achieved in terms of portfolio variance minimization.
To quantify these ideas, let P represent an overall portfolio consisting of m stocks where si(n) represents the ith stock price at time index n and ai>0 the weight factor associated with the ith stock. Note that the unit of time can be hours, days, months or years depending on the investment duration. Clearly
                                          a            i                    >          0                ,                                            ∑                              i                =                1                            m                        ⁢                          a              i                                =          1                                    (        1        )            and the ais are unknown to start with.
If Co represents the total capital, then Coai represents the capital invested in the ith stock so that Coai/si(0)=ki represents the actual number of the ith stock in the portfolio. Hence the portfolio value at time index n equals to
      ∑          i      =      1        m    ⁢            k      i        ⁢                  s        i            ⁡              (        n        )            and hence the portfolio return over duration (0, n) equals
                                                        P              =                                                                    ∑                                          i                      =                      1                                        m                                    ⁢                                                            k                      i                                        ⁢                                                                  s                        i                                            ⁡                                              (                        n                        )                                                                                            -                                  C                  o                                                                                                        =                                                C                  o                                ⁢                                                      ∑                                          i                      =                      1                                        m                                    ⁢                                                            a                      i                                        ⁢                                                                                                                        s                            i                                                    ⁡                                                      (                            n                            )                                                                          -                                                                              s                            i                                                    ⁡                                                      (                            0                            )                                                                                                                                                s                          i                                                ⁡                                                  (                          0                          )                                                                                                                                                                                            =                                                C                  o                                ⁢                                                      ∑                                          i                      =                      1                                        m                                    ⁢                                                            a                      i                                        ⁢                                                                  r                        i                                            ⁡                                              (                        n                        )                                                                                                                                                    (        2        )            where
                                          r            i                    ⁡                      (            n            )                          =                                                            s                i                            ⁡                              (                n                )                                      -                                          s                i                            ⁡                              (                0                )                                                                        s              i                        ⁡                          (              0              )                                                          (        3        )            represents the ith stock return over the duration (0, n). Thus for portfolio return analysis, the important variable is the stock return value ri(n) rather than the actual stock value si(n) itself.
Letμi=E{ri(n)}  (4)represent the mean value (expected value) of the ith stock return (see, in “Probability, Random Variables and Stochastic Processes,” Fourth Edition, A. Papoulis, and S. U. Pillai, McGraw-Hill Companies, New York, USA, 2001). The mean value μi can also be a good indicator about the future trend, where one hopes the stock will be based on company performance and other related parameters. One may need to predict μi based on all available data. The stock return values move around their mean values, the individual variations depending on the individual variance and related cross-correlations among other stocks.
The expected value of the portfolio return represents the net gain G of the portfolio. Thus the overall gain of the portfolio in (2) is given by (Co=1)
                    G        =                              E            ⁢                          {              P              }                                =                                    E              ⁢                              {                                                      ∑                                          i                      =                      1                                        m                                    ⁢                                                            a                      i                                        ⁢                                                                  r                        i                                            ⁡                                              (                        n                        )                                                                                            }                                      =                                                            ∑                                      i                    =                    1                                    m                                ⁢                                                      a                    i                                    ⁢                                      μ                    i                                                              =                                                                    a                    _                                    T                                ⁢                                  μ                  _                                                                                        (        5        )            wherea=[a1, a2, a3, . . . am]T  (6)r(n)=[r1(n), r2(n), r3(n), . . . rm(n)]T  (7)andμ=E{r(n)}=[μ1, μ2, μ3, . . . μm]T.  (8)Here E{.} stands for the expected or ensemble averaging operation as in (4). The overall risk of the portfolio is given by the variance of the portfolio return that equals
                                                                        σ                P                2                            =                            ⁢                              E                ⁢                                  {                                                            [                                              P                        -                                                  E                          ⁢                                                      {                            P                            }                                                                                              ]                                        2                                    }                                                                                                        =                            ⁢                              E                ⁢                                  {                                                                                                                                                                        a                              _                                                        T                                                    ⁡                                                      [                                                                                                                            r                                  _                                                                ⁡                                                                  (                                  n                                  )                                                                                            -                                                              μ                                _                                                                                      ]                                                                          ⁡                                                  [                                                                                                                    r                                _                                                            ⁡                                                              (                                n                                )                                                                                      -                                                          μ                              _                                                                                ]                                                                    T                                        ⁢                                          a                      _                                                        }                                                                                                        =                            ⁢                                                                    a                    _                                    T                                ⁢                E                ⁢                                  {                                                                                    [                                                                                                            r                              _                                                        ⁡                                                          (                              n                              )                                                                                -                                                      μ                            _                                                                          ]                                            ⁡                                              [                                                                                                            r                              _                                                        ⁡                                                          (                              n                              )                                                                                -                                                      μ                            _                                                                          ]                                                              T                                    }                                ⁢                                  a                  _                                                                                                        =                            ⁢                                                                    a                    _                                    T                                ⁢                R                ⁢                                  a                  _                                                                                        (        9        )            where (see, in “Probability, Random Variables and Stochastic Processes,” Fourth Edition, A. Papoulis, and S. U. Pillai, McGraw-Hill Companies, New York, USA, 2001).R=E{[r(n)−μ][r(n)−μ]T}>0  (10)represents the covariance matrix (positive definite matrix) of the stock return vector r(n). Notice thatRii=E{(ri(n)−μi)2}=var{ri(n)}=σi2>0  (11)represents the variance of the ith stock return, andRij=E{(ri(n)−μi)(rj(n)−μj)}=cov{ri(n), rj(n)}=ρijσiσj  (12)represents the covariance between returns ri(n) and rj(n), where ρij is defined as the correlation coefficient between ri(n) and rj(n).
The above equations are well known in the prior art. In addition, the above equations have been used to formulate the following portfolio optimization strategy, which can be called “Prior Art: Minimize Portfolio Risk”.
Prior Art: Minimize Portfolio Risk:
Find the right max of stocks that minimizes the overall portfolio risk. Take whatever profit you get.
In the “Minimize Portfolio Risk” approach, the Portfolio risk is minimized by minimizing the portfolio variance σP2 in equation (9) subject to the constraints in equation (1). This gives the well-known constrained optimization problem referred to in “Mean-Variance Analysis in Portfolio Choice and Capital Markets”, H. M. Markowitz, et. al., John Willy, New York, 2000:min aTR asubject to aTe=1  (13)where e represents the “all ones” column vectore=[1, 1, 1, . . . , . . . 1]T  (14)Notice that the nonnegative constraint for a needs to be incorporated as well. One approach of the prior art is to use the simplex type optimization methods to incorporate the positivity constraint for the weight vector a as referred to in “Mean-Variance Analysis in Portfolio Choice and Capital Markets”, H. M. Markowitz, et. al. Another approach is to put additional constraints on stock selection to realize this goal.
Eq. (13) leads to the modified Lagrangian functionmin Λ=aTRa+λ(aTe−1)  (15)and its minimization yields
                                          ∂            Λ                                ∂                          a              _                                      =                                            2              ⁢              R              ⁢                              a                _                                      +                          λ              ⁢                                                          ⁢                              e                _                                              =          0                                    (        16        )            which gives
                              a          _                =                              -                          λ              2                                ⁢                      R                          -              1                                ⁢                      e            _                                              (        17        )            and the normalization condition
                    a        _            T        ⁢          e      _        =                    1        ⁢                    -              λ        2              =          1                                    e            _                    T                ⁢                  R                      -            1                          ⁢                  e          _                    or
                              a          _                =                                                            R                                  -                  1                                            ⁢                              e                _                                                                                      e                  _                                T                            ⁢                              R                                  -                  1                                            ⁢                              e                _                                              >          0.                                    (        18        )            Observe that (18) must turn out to be a positive vector. This is clearly satisfied if R−1 is a positive (Perron) matrix as specified in the prior art in “Matrix Algebra and Its Applications for Statistics and Econometrics”, C. R. Rao, M. B. Rao, Singapore,: World Scientific, 1998. Thus if R−1 is a positive definite matrix, then the optimum vector a turns out to be positive since the denominator eTR−1e>0.
In other words, to start with one may select only those stocks to be in the portfolio for which R−1 satisfies the Perron property (positive matrix). In that case, the minimum volatility is given by:
                                          (                          σ              P              2                        )                    min                =                                                            a                _                            T                        ⁢                          R                              -                1                                      ⁢                          a              _                                =                                    1                                                                    e                    _                                    T                                ⁢                                  R                                      -                    1                                                  ⁢                                  e                  _                                                      =                                          1                                                      ∑                    i                                    ⁢                                                            ∑                      j                                        ⁢                                          R                      ij                                                                                  >              0.                                                          (        19        )            where Rij represents the (i,j)-th entry of the matrix R−1.Also, the net gain in that case is given by
                    G        =                                                            a                _                            T                        ⁢                          μ              _                                =                                                                                          e                    _                                    T                                ⁢                                  R                                      -                    1                                                  ⁢                                  μ                  _                                                                                                  e                    _                                    T                                ⁢                                  R                                      -                    1                                                  ⁢                                  e                  _                                                      >            0.                                              (        20        )            
For example, in a two-stock portfolio, the Perron property that R−1 contain only positive entries is satisfied by any two negatively correlated stocks since in that case
                              R          =                      (                                                            1                                                                      -                    ρ                                                                                                                    -                    ρ                                                                    1                                                      )                          ,                  0          <          ρ          <          1                                    (        21        )            and
                              R                      -            1                          =                                            1                              1                -                                  ρ                  2                                                      ⁢                          (                                                                    1                                                        ρ                                                                                        ρ                                                        1                                                              )                                >          0                                    (        22        )            has all positive entries. Observe that equation (21) represents the covariance matrix of two stock returns with “opposing trends” and they are negatively correlated. Hence when one “goes up”, the tendency of the other one is to “go down” thus minimizing the risk of loss. For large m, realizing this nonnegativity condition may be too restrictive. From equation (18), a more relaxed condition is that the row sums of R−1 must be all positive.
From time-to-time, the portfolio manager should recompute R and update the portfolio mix vector a by buying/selling stocks to keep the overall portfolio volatility low.